American Institute of Mathematical Sciences

Advanced
February  2019, 24(2): 831-849. doi: 10.3934/dcdsb.2018209

Boundedness in a three-dimensional Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation

Dan Li 1,, , Chunlai Mu 1, , Pan Zheng 2, and  Ke Lin 3,

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
2. 

Department of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
3. 

College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

* Corresponding author: liwadanj@126.com

Received  October 2017 Revised  March 2018 Published  June 2018

Fund Project: The first author is partially supported by graduate research and innovation foundation of Chongqing, China (Grant No. CYB 17040). The second author is partially supported by NSFC (Grant Nos. 11771062, 11571062), the Basic and Advanced Research Project of CQC-STC (Grant No. cstc2015jcyjBX0007) and Fundamental Research Funds for the Central Universities (Grant Nos. 10611CDJXZ238826). The third author is partially supported by National Science Foundation of China (Grant Nos. 11601053, 11526042). The fourth author is partially supported by the Fundamental Research Funds for the Central Universities (Grant No. JBK1801059).

This paper deals with a boundary-value problem for a coupled chemotaxis-Stokes system with logistic source
$\begin{eqnarray*}\left\{\begin{array}{llll}n_t+u·\nabla n = \nabla·(D(n)\nabla n)-\nabla·(n \mathcal{S}(x, n, c)·\nabla c)\\ +ξ n-μ n^{2}, &x∈ Ω, &t>0, \\c_{t}+u·\nabla c = Δ c-c+n, &x∈Ω, &t>0, \\u_{t}+\nabla P = Δ u+n\nablaφ, &x∈Ω, &t>0, \\\nabla· u = 0, &x∈Ω, &t>0\end{array}\right.\end{eqnarray*}$
in three-dimensional smoothly bounded domains, where the parameters $ξ\ge0$, $μ>0$ and $φ∈ W^{1, ∞}(Ω)$, $D$ is a given function satisfying $D(n)\ge C_{D}n^{m-1}$ for all $n>0$ with $m>0$ and $C_{D}>0$. $\mathcal{S}$ is a given function with values in $\mathbb{R}^{3×3}$ which fulfills
$ \begin{equation*}{\label{1.3}}\begin{split}|\mathcal{S}(x, n, c)|\leq C_{\mathcal{S}}(1+n)^{-α}\end{split}\end{equation*}$
with some $C_{\mathcal{S}}>0$ and $α>0$. It is proved that for all reasonably regular initial data, global weak solutions exist whenever $m+2α>\frac{6}{5}$. This extends a recent result by Liu el at. (J. Diff. Eqns, 261 (2016) 967-999) which asserts global existence of weak solutions under the constraints $m+α>\frac{6}{5}$ and $m\ge\frac{1}{3}$.
Keywords: Chemotaxis- Stokes, nonlinear diffusion, tensor-valued sensitivity, boundedness, logistic source.
Mathematics Subject Classification: Primary: 35K45; Secondary: 35B65, 35Q92, 35Q35, 92C17.
Citation: Dan Li, Chunlai Mu, Pan Zheng, Ke Lin. Boundedness in a three-dimensional Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 831-849. doi: 10.3934/dcdsb.2018209
References:
[1]

K. Baghaei and M. Hesaaraki, Global existence and boundedness of classical solutions for a chemotaxis model with logstic source, C. R. Acad. Sci. Paris. Ser. I, 351 (2013), 585-591.  doi: 10.1016/j.crma.2013.07.027.  Google Scholar

[2]

V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl., 86 (2006), 155-175.  doi: 10.1016/j.matpur.2006.04.002.  Google Scholar

[3]

X. Cao and S. Ishida, Global-in-time bounded weak solutions to a degenerate quasilinear Keller-Segel system with rotation, Nonlinearity, 27 (2014), 1899-1913.  doi: 10.1088/0951-7715/27/8/1899.  Google Scholar

[4]

T. Cieślak, Quasilinear nonuniformly parabolic system modelling chemotaxis, J. Math. Anal. Appl., 326 (2007), 1410-1426.  doi: 10.1016/j.jmaa.2006.03.080.  Google Scholar

[5]

T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045.  Google Scholar

[6]

T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasilinear system Keller-Segel system and applications to volume filling models, J. Differnetial Equations, 258 (2015), 2080-2113.  doi: 10.1016/j.jde.2014.12.004.  Google Scholar

[7]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemoatxis, Nonlinearity, 21 (2008), 1057-1076.  doi: 10.1088/0951-7715/21/5/009.  Google Scholar

[8]

K. Djie and M. Winkler, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045.  Google Scholar

[9]

K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684.  doi: 10.1016/j.jmaa.2014.11.045.  Google Scholar

[10]

M. Herrero and J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa Cl. Sci., 24 (1997), 633-683.   Google Scholar

[11]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[12]

T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 281-301.  doi: 10.1006/aama.2001.0721.  Google Scholar

[13]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.  Google Scholar

[14]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[15]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I. Jahresber. Dtsch. Math.- Ver., 105 (2003), 103-165.   Google Scholar

[16]

S. Ishida, Global existence and boundedness for chemotaxis-Navier-Stokes system with position-dependent sensitivity in 2d bounded domains, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 3463-3482.  doi: 10.3934/dcds.2015.35.3463.  Google Scholar

[17]

S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.  Google Scholar

[18]

S. Ishida and T. Yokota, Blow-up finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, Discrete Continuous Dynam. Systems - B, 18 (2013), 2569-2596.  doi: 10.3934/dcdsb.2013.18.2569.  Google Scholar

[19]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[20]

A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and efficiency of biological reactions: the critical reaction case, J. Math. Phys., 53 (2012), 115609, 9pp. doi: 10.1063/1.4742858.  Google Scholar

[21]

A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and enhancement of biological reactions, Comm. Partial Differential Equations, 37 (2012), 298-318.  doi: 10.1080/03605302.2011.589879.  Google Scholar

[22]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.  doi: 10.1016/j.jmaa.2008.01.005.  Google Scholar

[23]

H. A. Levine and B. D. Sleeman, A system of reaction-diffusion equations arising in the theory of reinforced randomw walks, SIAM J. Appl. Math., 57 (1997), 683-730.  doi: 10.1137/S0036139995291106.  Google Scholar

[24]

T. Li, A. Suen, M. Winkler and C. Xue, Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms, Math. Models Methods Appl. Sci., 25 (2015), 721-746.  doi: 10.1142/S0218202515500177.  Google Scholar

[25]

X. Li, Y. Wang and Z. Xiang, Global existence and boundedness in a 2D KellerSegel-Stokes system with nonlinear diffusion and rotational flux, Commun. Math. Sci., 14 (2016), 1889-1910.  doi: 10.4310/CMS.2016.v14.n7.a5.  Google Scholar

[26]

X. Li and Y. Xiao, Global existence and boundedness in a 2D Keller-Segel-Stokes system, Nonlinear Anal., 37 (2017), 14-30.  doi: 10.1016/j.nonrwa.2017.02.005.  Google Scholar

[27]

J. Liu and Y. Wang, Global existence and boundedness in a Keller-Segel- (Navier-)Stokes system with signal-dependent sensitivity, J. Math. Anal. Appl., 447 (2017), 499-528.  doi: 10.1016/j.jmaa.2016.10.028.  Google Scholar

[28]

J. Liu and Y. Wang, Boundedness and decay property in a three-dimensionlal Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation, J. Differential Equations, 261 (2016), 967-999.  doi: 10.1016/j.jde.2016.03.030.  Google Scholar

[29]

T. Nagai, Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.   Google Scholar

[30]

K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar

[31]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.   Google Scholar

[32]

Y. Peng and Z. Xiang, Global existence and boundedness in a 3D Keller-Segel-Stokes system with nonlinear diffusion and rotational flux, Zeitschrift f/"ur angewandte Mathematik und Physik, 68 (2017), p68. doi: 10.1007/s00033-017-0816-6.  Google Scholar

[33]

Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasilinear parabolic systems of chemotaxis, Differential Integral Equations, 20 (2007), 133-180.   Google Scholar

[34]

Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system, Zeitschrift f/"ur angewandte Mathematik und Physik, 67 (2016), p138. doi: 10.1007/s00033-016-0732-1.  Google Scholar

[35]

Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.  doi: 10.1007/s00033-015-0541-y.  Google Scholar

[36]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[37]

Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1901-1914.  doi: 10.3934/dcds.2012.32.1901.  Google Scholar

[38]

Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178.  doi: 10.1016/j.anihpc.2012.07.002.  Google Scholar

[39]

M. Winkler, Global weak solution in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352.  doi: 10.1016/j.anihpc.2015.05.002.  Google Scholar

[40]

I. Tuval, L. Cisneros and C. Dombrowski, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad, Sci. USA, 102 (2005), 2277-2282.  doi: 10.1073/pnas.0406724102.  Google Scholar

[41]

L. C. Wang, Y. Li and C. L. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 789-802.  doi: 10.3934/dcds.2014.34.789.  Google Scholar

[42]

L. Wang and C. L. Mu, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.  doi: 10.1016/j.jde.2013.12.007.  Google Scholar

[43]

Y. Wang and X. Cao, Global classical solutions of a 3d chemotaxis-Stokes system with rotation, Discrete Continuous Dynam. Systems - B, 20 (2015), 3235-3254.  doi: 10.3934/dcdsb.2015.20.3235.  Google Scholar

[44]

Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations, 259 (2015), 7578-7609.  doi: 10.1016/j.jde.2015.08.027.  Google Scholar

[45]

Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: the 3D case, J. Differential Equations, 261 (2016), 4944-4973.  doi: 10.1016/j.jde.2016.07.010.  Google Scholar

[46]

Y. Wang and L. Xie, Boundedness for a 3D chemotaxis-Stokes system with porous medium diffusion and tensor-valued chemotaxis sensitivity. Z. Angew. Math. Phys., 68 (2017), p29. doi: 10.1007/s00033-017-0773-0.  Google Scholar

[47]

Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system, Math. Models Methods Appl. Sci., 27 (2017), 2745-2780.  doi: 10.1142/S0218202517500579.  Google Scholar

[48]

M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations, 54 (2015), 3789-3828.  doi: 10.1007/s00526-015-0922-2.  Google Scholar

[49]

M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925.  doi: 10.1002/mma.319.  Google Scholar

[50]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.  Google Scholar

[51]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[52]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar

[53]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong dampening, J. Differential Equations, 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023.  Google Scholar

[54]

M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673.  doi: 10.1002/mana.200810838.  Google Scholar

[55]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

[56]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modelling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.  Google Scholar

[57]

M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352.  doi: 10.1016/j.anihpc.2015.05.002.  Google Scholar

[58]

M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487.  doi: 10.1007/s00205-013-0678-9.  Google Scholar

[59]

M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045.  Google Scholar

[60]

D. Wrzosek, Global attractor for a chemotaxis model with prevention of overcrowding, Nonlinear Anal., 59 (2004), 1293-1310.  doi: 10.1016/j.na.2004.08.015.  Google Scholar

[61]

C. Xue, Macroscopic equations for bacterial chemotaxis: Integration of detailed biochemistry of cell signalling, J. Math. Biol., 70 (2015), 1-44.  doi: 10.1007/s00285-013-0748-5.  Google Scholar

[62]

C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial population, SIAM J. Appl. Math, 70 (2009), 133-167.  doi: 10.1137/070711505.  Google Scholar

[63]

Q. Zhang and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 259 (2015), 3730-3754.  doi: 10.1016/j.jde.2015.05.012.  Google Scholar

[64]

J. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140.  doi: 10.1016/j.jde.2015.02.003.  Google Scholar

[65]

J. Zheng, Boundedness in a three-dimensional chemotaxis-fluid system involving tensor-valued sensitivity with saturation, J. Math. Anal. Appl., 442 (2016), 353-375.  doi: 10.1016/j.jmaa.2016.04.047.  Google Scholar

[66]

J. Zheng, Boundedness of solutions to a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 431 (2015), 867-888.  doi: 10.1016/j.jmaa.2015.05.071.  Google Scholar

[67]

J. Zheng, A new approach toward locally bounded global solutions to a 3D chemotaxis-Stokes system with nonlinear diffusion and rotation, preprint, arXiv: 1701.01334. Google Scholar

[68]

J. Zheng, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 263 (2017), 2606-2629.  doi: 10.1016/j.jde.2017.04.005.  Google Scholar

show all references

References:
[1]

K. Baghaei and M. Hesaaraki, Global existence and boundedness of classical solutions for a chemotaxis model with logstic source, C. R. Acad. Sci. Paris. Ser. I, 351 (2013), 585-591.  doi: 10.1016/j.crma.2013.07.027.  Google Scholar

[2]

V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl., 86 (2006), 155-175.  doi: 10.1016/j.matpur.2006.04.002.  Google Scholar

[3]

X. Cao and S. Ishida, Global-in-time bounded weak solutions to a degenerate quasilinear Keller-Segel system with rotation, Nonlinearity, 27 (2014), 1899-1913.  doi: 10.1088/0951-7715/27/8/1899.  Google Scholar

[4]

T. Cieślak, Quasilinear nonuniformly parabolic system modelling chemotaxis, J. Math. Anal. Appl., 326 (2007), 1410-1426.  doi: 10.1016/j.jmaa.2006.03.080.  Google Scholar

[5]

T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045.  Google Scholar

[6]

T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasilinear system Keller-Segel system and applications to volume filling models, J. Differnetial Equations, 258 (2015), 2080-2113.  doi: 10.1016/j.jde.2014.12.004.  Google Scholar

[7]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemoatxis, Nonlinearity, 21 (2008), 1057-1076.  doi: 10.1088/0951-7715/21/5/009.  Google Scholar

[8]

K. Djie and M. Winkler, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045.  Google Scholar

[9]

K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684.  doi: 10.1016/j.jmaa.2014.11.045.  Google Scholar

[10]

M. Herrero and J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa Cl. Sci., 24 (1997), 633-683.   Google Scholar

[11]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[12]

T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 281-301.  doi: 10.1006/aama.2001.0721.  Google Scholar

[13]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.  Google Scholar

[14]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[15]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I. Jahresber. Dtsch. Math.- Ver., 105 (2003), 103-165.   Google Scholar

[16]

S. Ishida, Global existence and boundedness for chemotaxis-Navier-Stokes system with position-dependent sensitivity in 2d bounded domains, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 3463-3482.  doi: 10.3934/dcds.2015.35.3463.  Google Scholar

[17]

S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.  Google Scholar

[18]

S. Ishida and T. Yokota, Blow-up finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, Discrete Continuous Dynam. Systems - B, 18 (2013), 2569-2596.  doi: 10.3934/dcdsb.2013.18.2569.  Google Scholar

[19]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[20]

A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and efficiency of biological reactions: the critical reaction case, J. Math. Phys., 53 (2012), 115609, 9pp. doi: 10.1063/1.4742858.  Google Scholar

[21]

A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and enhancement of biological reactions, Comm. Partial Differential Equations, 37 (2012), 298-318.  doi: 10.1080/03605302.2011.589879.  Google Scholar

[22]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.  doi: 10.1016/j.jmaa.2008.01.005.  Google Scholar

[23]

H. A. Levine and B. D. Sleeman, A system of reaction-diffusion equations arising in the theory of reinforced randomw walks, SIAM J. Appl. Math., 57 (1997), 683-730.  doi: 10.1137/S0036139995291106.  Google Scholar

[24]

T. Li, A. Suen, M. Winkler and C. Xue, Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms, Math. Models Methods Appl. Sci., 25 (2015), 721-746.  doi: 10.1142/S0218202515500177.  Google Scholar

[25]

X. Li, Y. Wang and Z. Xiang, Global existence and boundedness in a 2D KellerSegel-Stokes system with nonlinear diffusion and rotational flux, Commun. Math. Sci., 14 (2016), 1889-1910.  doi: 10.4310/CMS.2016.v14.n7.a5.  Google Scholar

[26]

X. Li and Y. Xiao, Global existence and boundedness in a 2D Keller-Segel-Stokes system, Nonlinear Anal., 37 (2017), 14-30.  doi: 10.1016/j.nonrwa.2017.02.005.  Google Scholar

[27]

J. Liu and Y. Wang, Global existence and boundedness in a Keller-Segel- (Navier-)Stokes system with signal-dependent sensitivity, J. Math. Anal. Appl., 447 (2017), 499-528.  doi: 10.1016/j.jmaa.2016.10.028.  Google Scholar

[28]

J. Liu and Y. Wang, Boundedness and decay property in a three-dimensionlal Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation, J. Differential Equations, 261 (2016), 967-999.  doi: 10.1016/j.jde.2016.03.030.  Google Scholar

[29]

T. Nagai, Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.   Google Scholar

[30]

K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar

[31]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.   Google Scholar

[32]

Y. Peng and Z. Xiang, Global existence and boundedness in a 3D Keller-Segel-Stokes system with nonlinear diffusion and rotational flux, Zeitschrift f/"ur angewandte Mathematik und Physik, 68 (2017), p68. doi: 10.1007/s00033-017-0816-6.  Google Scholar

[33]

Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasilinear parabolic systems of chemotaxis, Differential Integral Equations, 20 (2007), 133-180.   Google Scholar

[34]

Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system, Zeitschrift f/"ur angewandte Mathematik und Physik, 67 (2016), p138. doi: 10.1007/s00033-016-0732-1.  Google Scholar

[35]

Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.  doi: 10.1007/s00033-015-0541-y.  Google Scholar

[36]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[37]

Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1901-1914.  doi: 10.3934/dcds.2012.32.1901.  Google Scholar

[38]

Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178.  doi: 10.1016/j.anihpc.2012.07.002.  Google Scholar

[39]

M. Winkler, Global weak solution in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352.  doi: 10.1016/j.anihpc.2015.05.002.  Google Scholar

[40]

I. Tuval, L. Cisneros and C. Dombrowski, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad, Sci. USA, 102 (2005), 2277-2282.  doi: 10.1073/pnas.0406724102.  Google Scholar

[41]

L. C. Wang, Y. Li and C. L. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 789-802.  doi: 10.3934/dcds.2014.34.789.  Google Scholar

[42]

L. Wang and C. L. Mu, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.  doi: 10.1016/j.jde.2013.12.007.  Google Scholar

[43]

Y. Wang and X. Cao, Global classical solutions of a 3d chemotaxis-Stokes system with rotation, Discrete Continuous Dynam. Systems - B, 20 (2015), 3235-3254.  doi: 10.3934/dcdsb.2015.20.3235.  Google Scholar

[44]

Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations, 259 (2015), 7578-7609.  doi: 10.1016/j.jde.2015.08.027.  Google Scholar

[45]

Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: the 3D case, J. Differential Equations, 261 (2016), 4944-4973.  doi: 10.1016/j.jde.2016.07.010.  Google Scholar

[46]

Y. Wang and L. Xie, Boundedness for a 3D chemotaxis-Stokes system with porous medium diffusion and tensor-valued chemotaxis sensitivity. Z. Angew. Math. Phys., 68 (2017), p29. doi: 10.1007/s00033-017-0773-0.  Google Scholar

[47]

Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system, Math. Models Methods Appl. Sci., 27 (2017), 2745-2780.  doi: 10.1142/S0218202517500579.  Google Scholar

[48]

M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations, 54 (2015), 3789-3828.  doi: 10.1007/s00526-015-0922-2.  Google Scholar

[49]

M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925.  doi: 10.1002/mma.319.  Google Scholar

[50]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.  Google Scholar

[51]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[52]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar

[53]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong dampening, J. Differential Equations, 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023.  Google Scholar

[54]

M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673.  doi: 10.1002/mana.200810838.  Google Scholar

[55]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

[56]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modelling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.  Google Scholar

[57]

M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352.  doi: 10.1016/j.anihpc.2015.05.002.  Google Scholar

[58]

M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487.  doi: 10.1007/s00205-013-0678-9.  Google Scholar

[59]

M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045.  Google Scholar

[60]

D. Wrzosek, Global attractor for a chemotaxis model with prevention of overcrowding, Nonlinear Anal., 59 (2004), 1293-1310.  doi: 10.1016/j.na.2004.08.015.  Google Scholar

[61]

C. Xue, Macroscopic equations for bacterial chemotaxis: Integration of detailed biochemistry of cell signalling, J. Math. Biol., 70 (2015), 1-44.  doi: 10.1007/s00285-013-0748-5.  Google Scholar

[62]

C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial population, SIAM J. Appl. Math, 70 (2009), 133-167.  doi: 10.1137/070711505.  Google Scholar

[63]

Q. Zhang and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 259 (2015), 3730-3754.  doi: 10.1016/j.jde.2015.05.012.  Google Scholar

[64]

J. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140.  doi: 10.1016/j.jde.2015.02.003.  Google Scholar

[65]

J. Zheng, Boundedness in a three-dimensional chemotaxis-fluid system involving tensor-valued sensitivity with saturation, J. Math. Anal. Appl., 442 (2016), 353-375.  doi: 10.1016/j.jmaa.2016.04.047.  Google Scholar

[66]

J. Zheng, Boundedness of solutions to a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 431 (2015), 867-888.  doi: 10.1016/j.jmaa.2015.05.071.  Google Scholar

[67]

J. Zheng, A new approach toward locally bounded global solutions to a 3D chemotaxis-Stokes system with nonlinear diffusion and rotation, preprint, arXiv: 1701.01334. Google Scholar

[68]

J. Zheng, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 263 (2017), 2606-2629.  doi: 10.1016/j.jde.2017.04.005.  Google Scholar

[1]

Xie Li, Zhaoyin Xiang. Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3503-3531. doi: 10.3934/dcds.2015.35.3503
[2]

Qi Wang, Jingyue Yang, Feng Yu. Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 5021-5036. doi: 10.3934/dcds.2017216
[3]

Guoqiang Ren, Bin Liu. Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3843-3883. doi: 10.3934/cpaa.2020170
[4]

Liangchen Wang, Yuhuan Li, Chunlai Mu. Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 789-802. doi: 10.3934/dcds.2014.34.789
[5]

Ling Liu, Jiashan Zheng. Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3357-3377. doi: 10.3934/dcdsb.2018324
[6]

Pan Zheng, Chunlai Mu, Xuegang Hu. Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2299-2323. doi: 10.3934/dcds.2015.35.2299
[7]

Shijie Shi, Zhengrong Liu, Hai-Yang Jin. Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source. Kinetic & Related Models, 2017, 10 (3) : 855-878. doi: 10.3934/krm.2017034
[8]

Pan Zheng, Chunlai Mu, Xiaojun Song. On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1737-1757. doi: 10.3934/dcds.2016.36.1737
[9]

Sachiko Ishida. Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3463-3482. doi: 10.3934/dcds.2015.35.3463
[10]

Hong Yi, Chunlai Mu, Guangyu Xu, Pan Dai. A blow-up result for the chemotaxis system with nonlinear signal production and logistic source. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020194
[11]

Tomomi Yokota, Noriaki Yoshino. Existence of solutions to chemotaxis dynamics with logistic source. Conference Publications, 2015, 2015 (special) : 1125-1133. doi: 10.3934/proc.2015.1125
[12]

Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617
[13]

Hai-Yang Jin. Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3595-3616. doi: 10.3934/dcds.2018155
[14]

Wei Wang, Yan Li, Hao Yu. Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3663-3669. doi: 10.3934/dcdsb.2017147
[15]

Pan Zheng. Global boundedness and decay for a multi-dimensional chemotaxis-haptotaxis system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 2039-2056. doi: 10.3934/dcdsb.2016035
[16]

Jiashan Zheng. Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 627-643. doi: 10.3934/dcds.2017026
[17]

Jie Zhao. Large time behavior of solution to quasilinear chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1737-1755. doi: 10.3934/dcds.2020091
[18]

Abelardo Duarte-Rodríguez, Lucas C. F. Ferreira, Élder J. Villamizar-Roa. Global existence for an attraction-repulsion chemotaxis fluid model with logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 423-447. doi: 10.3934/dcdsb.2018180
[19]

Ke Lin, Chunlai Mu. Global dynamics in a fully parabolic chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5025-5046. doi: 10.3934/dcds.2016018
[20]

Wenji Zhang, Pengcheng Niu. Asymptotics in a two-species chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020288

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (166)
  • HTML views (357)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences